TY - JOUR

T1 - Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors

AU - Zhou, Fujun

AU - Escher, Joachim

AU - Cui, Shangbin

N1 - Funding information: This work on the parts of the first and the third authors is supported by China National Science Foundation under the grant numbers 10471157 and 10771223. The first author also wishes to acknowledge his sincere thanks to the faculty and staff of the Institute for Applied Mathematics of the Leibniz University of Hannover for their hospitality during his visit there under the support of the KaiSi Foundation of Sun Yat-Sen University. We would like to thank the referee very much for the valuable suggestions.

PY - 2008/6/1

Y1 - 2008/6/1

N2 - In this paper we study well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors under the action of external inhibitors. An important feature of this problem is that the surface tension of the free boundary is taken into account. We first reduce this free boundary problem into an evolution equation in little Hölder space and use the well-posedness theory for differential equations in Banach spaces of parabolic type (i.e., equations which are treatable by using the analytic semi-group theory) to prove that this free boundary problem is locally well-posed for initial data belonging to a little Hölder space. Next we study flat solutions of this problem. We obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations. Finally we investigate asymptotic stability of flat stationary solutions under non-flat perturbations. By carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in Banach spaces of parabolic type, we give a complete analysis of stability and instability of all flat stationary solutions under small non-flat perturbations.

AB - In this paper we study well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors under the action of external inhibitors. An important feature of this problem is that the surface tension of the free boundary is taken into account. We first reduce this free boundary problem into an evolution equation in little Hölder space and use the well-posedness theory for differential equations in Banach spaces of parabolic type (i.e., equations which are treatable by using the analytic semi-group theory) to prove that this free boundary problem is locally well-posed for initial data belonging to a little Hölder space. Next we study flat solutions of this problem. We obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations. Finally we investigate asymptotic stability of flat stationary solutions under non-flat perturbations. By carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in Banach spaces of parabolic type, we give a complete analysis of stability and instability of all flat stationary solutions under small non-flat perturbations.

KW - Free boundary problem

KW - Inhibitors

KW - Multi-layer tumors

KW - Stability

KW - Well-posedness

UR - http://www.scopus.com/inward/record.url?scp=41949134707&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.02.038

DO - 10.1016/j.jde.2008.02.038

M3 - Article

AN - SCOPUS:41949134707

VL - 244

SP - 2909

EP - 2933

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -